I. The Introduction For The Quintessence of Quadratics unit, we took our knowledge about area, volume, and the Pythagorean Theorem and applied it to learning about quadratic equations, kinematics, and economics. From geometry to algebra, we were able to formulate rules and equations that could be applicable to our future learning about parabolas, standard form, and vertex form. Our initial problem was one involving kinematics. It stated that the students at HTHNC would be celebrating the end of the year with fireworks attached to a rocket. The three main questions to find an answer for were, “What is the maximum height of the rocket? When does the rocket reach its maximum height? And how long is the rocket in the air?” In order to start to answer these questions, we started by making the rocket’s height into a function of time, using the equation of h(t)=d0+v0·t+(1/2) a·t^2, which is written as y=ax^2+bx+c in quadratic form. d0 is the is the initial distance, v0 is the initial velocity, and a is the acceleration of the equation. The only force acting upon the rocket is gravity, making our next equation a=g=-31 ft/s^2. The acceleration would be negative because gravity’s force is going in the opposite direction.
Next, we looked at areas, using the corral variation problem. The questions asked us what the area of the initial corral was and how the area can be different if we change the length and width while keeping the total perimeter at 500 feet. By creating a table, we are able to see the changes:
For which we can make the equation of a(x)=x(500-2x) as the l*w=a We can then address that both the corral variation problem and the rocket problem are quadratic equations. By starting off with these types of problems, we were able to successfully transition into more challenging work.
II. Exploring the Vertex Form of the Quadratic Equation
A parabola’s equation can be written in Vertex Form. y = a (x - h)^2+k is the equation used for finding Vertex Form. In order for us to get a better understanding, we broke down the equation and looked at each variable individually.
The Coefficient of a Quadratic (a)
The value of a in this equation is affecting the width of the parabola. If the value of a is negative, the parabola opens downward and if it is positive the parabola opens up. The further a is from 0, the more narrow the parabola.
Y Coordinate of the Vertex (k)
k is the Y intercept of the vertex. When k is greater than 0, the plane becomes positive, and when it is less than 0, it is a negative plane. k determines the Y coordinate of the vertex.
X Coordinate of the Vertex (h)
h is the X intercept of the vertex. The parabola is affected when the X intercept is a positive or negative number to either concave up or down.
III. Other Forms of the Quadratic Equation
Standard Form of a Quadratic Equation
The standard form of a quadratic equation is ax^2+bx+c=0. a, b andc are known values. a cannot be equivalent to 0. x is the variable which we don't know yet. In the instance below, a is equal to 6, b is 5, and c is 2. This parabola is positive because it is concaving upward.
Vertex Form of a Quadratic Equation
The vertex form of a quadratic equation is y=a(x-h)^2+k. The vertex is at (h, k). The value of a determines whether or not the parabola is concaving up or down, meaning that depending on if a is positive or negative, the parabola will either go up or down.
Factored Form of a Quadratic Equation
Factored form is a(x - p)(x - q) = y. This form helps because it shows the x-intercepts clearly. If y = 4(x - 5)(x - 7), the x-intercept coordinate has a y value of 0. We substitute y for 0: 0 = 4(x - 5)(x - 7). We then know that one of the parenthesis is equal to 0 and makes it x = 4 and 6.
IV. Converting between Forms
Vertex to Standard- y=a(x-h)^2+kto standard form expand the equation to get rid of the parenthesis. Use the FOIL method --(x-h)^2 term and multiplying by a. Combine like-terms you get the standard form y=ax^2+bx+c
Vertex to Standard using an area diagram
Standard to Vertex Form- y=ax^2+bx+c to y=a(x-h)^2+k. You don't worry about the c term and start by factoring the a coefficient from ax^2+bx. Your equation should look like y=a(x^2+bx+d-d)+c. x^2+bx+d can be written as (x-h)^2. By multiplying the -d by a then adding it to c, will help you get the value of k. Then it is in vertex form y=a(x-h)^2+k
Factored to Standard Form- Start by using FOIL for binomials. For an equation y=(a+b)(c+d) you do the First, Outer, Inner, Last and end up with y=ac+ad+bc+bd. Using quadratics would look like a y=a(x-p)(x-q) and you end up with y=ax^2-axq-axp+apq
Standard to Factored Form- (-b+-√(b^2-4ac))/2a. a, b and c are the same constants from y=ax^2+bx+c, which equals the x intercepts of a parabola.
V. Solving Problems with Quadratic Equations
We were given problems that involved quadratic equations. This was through kinematics, geometry, and economic problems. This was for us to see how quadratics can be applicable outside of the classroom setting.
Kinematic Equation
Going back to our initial problem of this unit, The Rocket Launch, we can now use our new knowledge to finish solving the problem. Again, starting with the equation h(t)=-16t^2+92t+160. We learned the function is a parabola. To simplify the equation, we try to find the coordinates of the vertex and the positive x-intercept. By converting to vertex form to get the vertex, and factored form for the x intercept, we are able to get the equation to be h(t)=-16(t-2.875)^2+292.25. Maximum height of the rocket - 292.25 Amount of passed time until maximum height - 2.875 X intercept - 7.14883
Geometry and Economics
For Geometry, the "Leslie's Flowers" worksheet was most helpful. The problem stated that there was a gardener who was planning the layout of a new garden box. We used triangular structures and area rules to solve it. where we were tasked with helping a gardener plan the layout of her garden box, given that she used triangular structures. To incorporate Economics, the handout called "Widgets" gave us the task of solving for predicated sales of a company. We also had to solve for the gained profit, if x was the price for the company selling widgets.
Solving a Problem
"Is it a Homer?" was the 10th worksheet in the Quadratics unit. The problem states that when a baseball is thrown or hit, the air path is almost a perfect parabola. When Mighty Casey hits it, it hits a maximum height of 80 feet and is 200 feet away from the home base. The center fence is 380 feet from the home plate and is 15 feet tall. But does Mighty Casey's ball clear the fence?
We start with the equation y=a(x-h)^2+k. We can plug in 200 for the value of h and 80 for the value of k, making our new equation y=a(x-200)^2+80. We fill in 0 for x and y, giving us 0=a(0-200)^2+80. Then we subtract 80 from both sides and we are left with -80=a(-200)^2 and we divide both sides by -200^2 and we are left with our final answer of y=500(x-200)^2+80 after putting it back into vertex form. In conclusion, it clears!
VI. Reflection
It would be believed that I would have comprehended the forms, functions, and notations of quadratic expressions, I still do not fully understand how to calculate nor execute the transitions between vertex, standard, and factored forms of the equations. This math unit was not only rushed, but not in depthly explained. I feel as though we got worksheets on top of worksheets with multiple due dates and deadlines to have things completed that some were unable to actually understand what was being taught. Some may say to just ask questions, but due to incidents in class, I feel like I’ve shied away from asking, because other students tease. There are always things I can do to get help and I will admit I didn't take full advantage of that. In the upcoming Junior year, I am very nervous as I feel I am not ready for the concepts to come. The backbone of the 11th grade math skills come from Sophomore year. I can't say that I've been taught everything or understood what was attempted to be taught. I’m very nervous about college academics and the ACT. For the upcoming year, I hope that we can focus more on the ACT as a whole, as opposed to individual concepts for long chunks of time. In conclusion, for my Sophomore year math, I think it could have been taught with more explanation that was applicable all students learning styles.
Habits of a Mathematician
Looking for patterns-I started seeing similarities between linear functions and quadratic functions with some of the equations like y=mx+b and y=ax^2+bx+c Starting small- There were a lot of confusing concepts in this unit, so I knew that if I were to take it one problem at a time, I could get it done faster because I wouldn't be overwhelmed with all the other problems. Being systematic- By using the proper equations for different sections, I was able to complete the assigned tasks accurately and on time. I knew that I had to take a step back and highlight what equations would be used and where. Taking apart and putting back together- I would keep the problems separate in order to really be able to break things down. I looked at other worksheets in order to make sure I was on the right track. Conjecturing and testing- Every so often there would be a problem I would look at and now understand it at all. I would take the things I learned and use an estimation initially, and eventually would be able to come to an accurate conclusion. Staying organized- I knew that I had to keep all my work clear and concise in order to be able to obtain what was written. I tried to keep all my notes broken down into categories. That would help me be able to quickly look back and find what I was looking for Describing and articulating- I learned the importance of having a description of how you got to an answer, not just the answer itself because then if there were mistakes, it was easy to look back on the work and find where the mistake had been made and fixing it. seeking why and proving-Showing how I got to a conclusion was really beneficial to me so I could apply that the future problems because I would just repeat most of the same steps. Being confident, persistent, and patient- I was constantly getting frustrated with problems because I didn't understand how to solve them. It was really hard for me to be able to move from one problem to another if I felt like I didn't completely understand it. It made me feel really low. But, by taking my time with each problem and asking for help when needed, pushed me through the hard parts. Collaborating and listening- This came to my benefit a lot because in the beginning I had a really hard time understanding how to convert all the formulas. With the help from my peers, I was able to come to a better understanding because I got to see it through a different perspective. Generalizing- I learned that generalizing problems helped a lot because equations became applicable to other problems. I saw this especially in the standard, vertex, and factored form problems.